Irrational
numbers

The
procedure of
computing directions and drawing a figure can be generalised to
(positive)
irrational numbers. However, the problem is then that these figures
have an
infinite size. So, a figure can only be drawn partly after computing a
finite
number of directions.

The formula
is
similar to the rational numbers: N = K * D + E , where N, D and E are
positive
irrational numbers, K is a positive integer, and 0 < E <
D . In principle
D < N ; bigger values of D can be transformed just as for the
rational
numbers:

- Values of D in the range N < D < 2*N can be transformed to D' = 2*N-D; the figure based on D is mirrored with respect to the figure based on D' (apart from the first direction).
- Values of D bigger than 2*N can be transformed to D' = D mod (2*N)

The rules
for
reduction are similar as for the rational numbers.

A figure can
be
reduced to remove the "noise" of the higher generations. Because of
the irrational numbers a reduction process newer ends. A figure based
on
irrational numbers can be approximated for gaining insight, as follows.
Reduce
the figure for a few generations, and store the values of K and s .
Replace the
(reduced) values of N and D at the lowest generation by resembling
integers. And
then build up the higher generations again using the stored values of K
and s.
For example the pair (π, 1) can be approximated by (3230883,
1028422) .

Because
of the
usage of irrational numbers we
can divide both sides of the characteristic equation
by the same value without any further consequences. An advantage for
doing such
a division is to scale the numbers to convenient values. The equation
can
always be scaled such that D becomes 1 . In that situation, we define
the
numbers as being normalised.